2 n + 2 n + 1 + 2 n + 2 + ⋯ + 2 2 n = 2 2 n + 1 − 2 n {\displaystyle 2^{n}+2^{n+1}+2^{n+2}+\cdots +2^{2n}=2^{2n+1}-2^{n}}
L : 2 2 n = 2 2 → 2 + 2 2 = 6 R : 2 2 n + 1 − 2 n = 2 3 − 2 = 6 6 = 6 {\displaystyle {\begin{aligned}&L:2^{2n}=2^{2}\rightarrow 2+2^{2}=6\\&R:2^{2n+1}-2^{n}=2^{3}-2=6&6=6\\\end{aligned}}}
2 k + 2 k + 1 + 2 k + 2 + ⋯ + 2 2 k = 2 2 k + 1 − 2 k {\displaystyle 2^{k}+2^{k+1}+2^{k+2}+\cdots +2^{2k}=2^{2k+1}-2^{k}}
2 k + 1 + 2 k + 2 + ⋯ + 2 2 k ⏟ = 2 2 k + 1 − 2 k − 2 k + 2 2 k + 1 + 2 2 k + 2 = 2 2 k + 3 − 2 k + 1 2 2 k + 1 − 2 k − 2 k + 2 2 k + 1 + 2 2 k + 2 = 2 2 k + 3 − 2 k + 1 2 ∗ 2 2 k + 1 − 2 ∗ 2 k + 2 2 k + 2 = 2 2 k + 3 − 2 k + 1 2 2 ∗ 2 2 k − 2 ∗ 2 k + 2 2 k ∗ 2 2 = 2 2 k ∗ 2 3 − 2 k ∗ 2 / : 2 ∗ 2 k 2 ∗ 2 k − 1 + 2 k 2 = 2 k ∗ 2 2 − 1 2 k ∗ 2 2 = 2 k ∗ 2 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {2^{k+1}+2^{k+2}+\cdots +2^{2k}} _{=2^{2k+1}-2^{k}-2^{k}}+2^{2k+1}+2^{2k+2}=2^{2k+3}-2^{k+1}\\&{\color {blue}2^{2k+1}}{\color {green}-2^{k}-2^{k}}{\color {blue}+2^{2k+1}}+2^{2k+2}=2^{2k+3}-2^{k+1}\\&{\color {blue}2*2^{2k+1}}{\color {green}-2*2^{k}}+2^{2k+2}=2^{2k+3}-2^{k+1}\\&2^{2}*2^{2k}-2*2^{k}+2^{2k}*2^{2}=2^{2k}*2^{3}-2^{k}*2/:2*2^{k}\\&2*2^{k}-1+2^{k}2=2^{k}*2^{2}-1\\&2^{k}*2^{2}=2^{k}*2\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
